Sheikh Ibn Baz, a prominent Muslim scholar, says,

Zina (adultery and fornication) is one of the most serious of impermissible actions and the worst of major sins. Allah warns the disbelievers, murderers and adulterers of multiple punishments on the Day of Resurrection, as Allah says: 

(And those who invoke not any other ilah (god) along with Allah, nor kill such person as Allah has forbidden, except for just cause, nor commit illegal sexual intercourse — and whoever does this shall receive the punishment. The torment will be doubled to him on the Day of Resurrection, and he will abide therein in disgrace. Except those who repent and believe (in Islamic Monotheism), and do righteous deeds.” (Al-Furqan 25:68-70) 

The one who has fallen into such sin has to repent to Allah sincerely, and follow that with sincere belief and righteous deeds. Repentance is sincere if the penitent gives up the sin, regrets what has happened in the past and resolves never to go back to it, out of fear of Allah and awe of Him, hoping for His reward and fearing His punishment. Allah says: 

And verily, I am indeed forgiving to him who repents, believes (in My Oneness, and associates none in worship with Me) and does righteous good deeds, and then remains constant in doing them (till his death)” (Ta-Ha 20:82)

Both the man and the lady have to declare an unblemished repentance of that act and after having that, though they may consider the matter of marriage.

It should be noted here that a Muslim is not permitted to have a relation with a strange woman except through marriage with her. Unless her parents approve the marriage, they are not allowed to have a relationship with one another.

It’s important to understand that making a relation with a woman out of marriage is a sin which requires severe punishment from Allah, the Almighty. Thus, both the perpetrators are advised first to repent to Allah of what they have done, then if they still desire to marriage, the man can propose to her and try to make use of various permissible means to let her parents accept the marriage. If they agree, it is Ok; otherwise he must cut all his relations with her and start a new leaf of life, while observing the limits set by Islam.

However, all these rulings are true for this woman being a Muslim, if she is non-Muslim, then the case is much different. She must be a Christian or a Jew for it to be permissible to marry her, though it is a big risk to marry to a Christian or a Jewish woman. Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada , states:

“Although religiously speaking, it is permitted for Muslims to marry women belonging to the People of the Book (i.e., the Christians and Jews), this permission cannot be generalized. Even during the time of the second Caliph, `Umar ibn al-Khattab, we read that he forbade some of the eminent companions of the Prophet from marrying women of the People of the Book. He asked those companions: “If everyone were to make use of this permission, who would marry Muslim girls?”
For Caliph `Umar at that time it was only a question of Muslim women remaining unmarried. For us today, there are other complications arising out of such marriages.
Our experience with such marriages in North America (and other Western countries) compels us to conclude that after the initial phase of the honey-moon, often intractable problems may arise when the couple settles down to start living together and establishing a family. Such nagging issues include: Which religious festivals to celebrate; what type of food should be eaten, how are the children to be brought up? These are issues that pose serious challenges in marriage. It is not unusual to see that sometimes the father is even prevented from praying in front of his own children, while they are taken to church on a weekly basis. It is therefore not at all surprising when we see that the vast majority of such marriages end up in court.
the heavy toll of such marriages on the children cannot be over emphasized. The absence of a unified spiritual vision is bound to produce a generation of confused people who are totally deprived of any religious vision or ideals. Thus, in the final analysis, such marriages cost dearly spiritually, financially and emotionally.
If the man still wish to proceed with this marriage, then he must know that he is taking a big risk. Nobody can stop him from doing it, but he alone is responsible for his action.
Last but not least, the parents have every right to advise him against it. After all, they have brought him up and sacrificed for him, and so it is not wrong if they offer the best advice they can.”

May Allah guide all of us to what pleases Him!